And on a data connectivity change, e.g. GSM <-> WiFi switch, there is a good chance that a dozen other components start trying to re-establish their connections, which means the radio is awake anyway (for, usually, at most a few seconds).
An idle connection doesn't consume anything, but a useful idle connection will have some keepalive every X minutes. Multiply that by the number of connections your application will have and by the number of background application you run on your phone, and the radio will never truly sleep or enter the "low" mode. The solution wanted by Google and Apple is to maintain a single connection to their server on which every push is aggregated and sent, and that the OS itself manages.
> Multiply that by the number of connections your application will have and by the number of background application you run on your phone, and the radio will never truly sleep or enter the "low" mode.
Why would X x Y idle connections (X: TCP connections per app, Y: applictions) prevent your radio from sleeping?
In the worst case, if the pings aren't sent at the same moment and all applications have a timeout set to, let's say, 30 minutes, your phone will send a ping every 30/(X*Y) minutes. I don't know what the usual X and Y would be, but if they're high enough your radio will always be up (check out these slides, 23 in particular: https://www.igvita.com/slides/2013/io-radio-performance.pdf)
And that's only counting idle applications. If you have one that periodically polls a server, it gets even worse.